Spring 2023 THE SMALL FINITISTIC DIMENSIONS OF COMMUTATIVE RINGS
Xiaolei Zhang, Fanggui Wang
J. Commut. Algebra 15(1): 131-138 (Spring 2023). DOI: 10.1216/jca.2023.15.131

Abstract

Let R be a commutative ring with identity. The small finitistic dimension fPD(R) of R is defined to be the supremum of projective dimensions of R-modules with finite projective resolutions. In this paper, we characterize a ring R with fPD(R)n using finitely generated semiregular ideals, tilting modules, cotilting modules of cofinite type and vaguely associated prime ideals. As an application, we obtain that if R is a Noetherian ring, then fPD(R)=sup{grade(𝔪,R)|𝔪Max(R)} where grade(𝔪,R) is the grade of 𝔪 on R. We also show that a ring R satisfies fPD(R)1 if and only if R is a DW ring. As applications, we show that the small finitistic dimensions of strong Prüfer rings and LPVDs are at most one. Moreover, for any given n, we obtain a total ring of quotients R satisfying fPD(R)=n.

Citation

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Xiaolei Zhang. Fanggui Wang. "THE SMALL FINITISTIC DIMENSIONS OF COMMUTATIVE RINGS." J. Commut. Algebra 15 (1) 131 - 138, Spring 2023. https://doi.org/10.1216/jca.2023.15.131

Information

Received: 16 July 2021; Revised: 6 November 2021; Accepted: 8 November 2021; Published: Spring 2023
First available in Project Euclid: 20 June 2023

MathSciNet: MR4604791
zbMATH: 07725180
Digital Object Identifier: 10.1216/jca.2023.15.131

Subjects:
Primary: 13D05 , 13D30

Keywords: DW ring , Noetherian ring , Prüfer ring , small finitistic dimension , tilting module

Rights: Copyright © 2023 Rocky Mountain Mathematics Consortium

Vol.15 • No. 1 • Spring 2023
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